10th

World Congress on Structural and Multidisciplinary Optimization

May 19 -24, 2013, Orlando, Florida, USA

1

Maximum member sizes and multiple concurrent optimization paths within a binary

material topology optimization method

Christian Brecher1, Simo Schmidt

2, Sierk Fiebig

3

1 WZL at RWTH Aachen University, Aachen, Germany, c.brecher@wzl.rwth-aachen.de

2 WZL at RWTH Aachen University, Aachen, Germany, simo.schmidt@wzl.rwth-aachen.de 3 Volkswagen AG, Brunswick, Germany, sierk.fiebig@volkswagen.de

1. Abstract

The constant increase in productivity and manufacturing efficiency drives the industry to incorporate optimization

techniques beyond the capabilities of manual optimization and numerous iterative loops within the product design

phase. As a result, topology optimization is becoming an integral part of the design process. Nevertheless, design

proposals obtained from topology optimization do not always meet all manufacturing constraints, and can be

difficult to interpret. Usually, these proposals have to be modified more or less extensively by experienced

engineers to be viable for prototyping and production. Oftentimes, these manual modifications and redesigns

adversely affect the optimality of the design proposal.

This problem is particularly evident for cast parts. One of the reasons for poor castability is the occurrence of

substructures with large cross-sectional areas connected to thin struts, large pockets of material, and other

discontinuities in wall thicknesses within the part, all of them increasing the risk of casting defect formation. Some

commercial topology optimization tools already allow for the consideration of maximum member sizes. This

restriction aims to decrease the possible range of wall thicknesses that can occur within the design proposal. The

implementation in mathematical optimization methods is fairly unproblematic. For heuristic iterative optimization

methods, which might start with a full design space and huge member sizes, the matter becomes rather difficult.

Using a new binary material topology optimization method developed at the Volkswagen AG as a starting point, a

mechanism was designed to enforce maximum member sizes via repetitive discrete events within the optimization

process. These act as a penalization of substructures with maximum member sizes exceeding a specified limit, but

allow for the structure to be repaired by the optimizer between these events. The modified optimization method

was successfully tested for robustness and convergence on two- and three-dimensional academic test problems and

industrial parts. In direct comparison to a commercial topology optimization product, the obtained results were

promising. To increase the coverage of the possible solution space, a mechanism for optimization branching was

developed. The possibility to simultaneously pursue both an unaltered and modified version of the structure at each

discrete penalization event generates a binary tree of variations which have been shown to converge in several

different local optima with a high degree of optimality, considering the additional member size restriction. This

work was done to provide a framework for further development of both maximum member size restrictions and

branching optimizations embedded in a heuristic iterative optimization process.

2. Keywords: industrial topology optimization, concurrent optimization paths, casting restrictions, discrete

penalization, maximum member size

3. Introduction

The field of topology optimization utilizing the Finite Element Method (FEM) has evolved rapidly in the past

decades. Fundamental work includes the development of mathematical optimization techniques in conjunction

with material models like the microstructure homogenization approach by Bendsoe and Kikuchi (1988) and the

popular Solid Isoptropic Material with Penalization (SIMP) model (Bendsoe, Zhou, Rozvany, 1989-1991). The

introduction of a heuristic structural optimization method presented by Mattheck et. al. (1992) proposed an entirely

different approach to the rigorous mathematical techniques, using heuristical growth and decay mechanisms

inspired by nature instead. Since then, a vast array of different optimization techniques was proposed. A review of

these developments can be found in Eschenauer and Olhoff (2001)[1], for example. Due to the heuristic nature of

some of these alternative optimization methods, concerns were raised as to the efficiency and usefulness of those

novel approaches (Rozvany, 2009, Sigmund, 2012). Nevertheless, some methods were accepted in industrial

applications (e.g. Harzheim and Graf, 1995, and successively TopShape, 2006, by the Adam Opel AG) due to the

ease of implementation, flexibility and low computational effort to achieve good, albeit not necessarily very close

to optimal, results [2].

Even though the acceptance of topology optimization methods within the industrial product development and

optimization cycle is constantly increasing, one of the disadvantages of all basic topology optimization methods is

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the lack of consideration for manufacturing constraints, resulting in design proposals that are close to optimal but

more or less useless since they cannot be produced economically [3]. With increasing maturity of topology

optimization methods, manufacturing constraints like casting or extrusion restrictions, prescribed symmetries and

patterns were integrated into leading commercial structural optimization software packages like Tosca® and

OptiStruct™, both of which utilize SIMP-based mathematical optimization techniques [4][5][6].

Due to the complexity of the casting process, casting restrictions in terms of pull directions (the direction, in which

the casting dies are separated) alone do not always suffice to guarantee good castability. To address this problem,

the authors propose the consideration of maximum member sizes within the optimization process to reduce

geometric discontinuities like large lumps of material adjacent to thin connecting struts in the resulting design

proposals, thus improving the overall castability of the optimized structures.

The inclusion of member size restrictions into the optimization problem is in itself not a new concept and was

investigated in a few research papers, for example by Guest (2008) [7] who proposed a radius-based constraint to

on local cross-sectional diameters in conjunction with a SIMP-based mathematical optimization approach. The

consideration of maximum member sizes in commercial topology optimization products was not supported up

until recently (e.g. outlook in Zhou et. al., 2002)[4], and is still deemed an experimental option within

OptiStruct™, for example [6]. The authors could not find any papers on evolutionary or heuristic iterative

topology optimization methods which consider maximum member sizes.

In this paper, a method to consider maximum member sizes within a heuristic iterative topology optimization

method using a binary material model is presented and compared to a mathematical approach used in OptiStruct™.

Furthermore, a branching technique to allow for the concurrent optimization of multiple structural variants is

proposed.

4. The binary material model approach

The methods and techniques presented in this paper were implemented as an extension of a topology optimization

method developed at the Volkswagen AG [8][9]. This optimization method utilizes a binary material model and

operates on a regular Cartesian grid of FE elements (often called voxels). It contains aspects of heuristic iterative

procedures like BESO (Querin, Steven, Xie, 2000), where FE elements within the design space are activated

and/or deactivated based on their level of stress (or other sensitivities) to successively generate lightweight design

proposals. Additionally, a controller mechanism dynamically adapts the amount of removed and added material in

each iteration to the development of the applied optimization constraints. The optimizer is connected to the

necessary FE analysis and system response postprocessing through a slim interface, Fig. 1(a).

without

casting

restriction

with

casting

restriction

(a) overview of the optimizer and its environment (b) the casting

restriction principle

visible elements (teal)

step size controller

basic rate

reduction

rate

correction

rateAdding elements

removing

elements

Structure

connected

Heuristics(e.g. deleting

unconnected elements)

no

yes

Optimizer

FEA analysisresult

postprocessing

simulation

Write Out

FEA model

Read in

resultsInterface

pull direction

Fig. 1: Basics of the binary material method. Source: [9]

In each iteration, the system response in terms of sensitivities and normalized constraint values is read in by the

optimizer. Here, the maximum of all normalized constraints is termed the normalized constraint value φ. It is

defined to be equal to 1 if the limit value is reached. Values above 1 correspond to a constraint violation. The

binary material method is capable of repairing infeasible structures by adding material to highly stressed regions

(called hotspots in engineering terminology) of the structure, as long as the available design space suffices.

3

Another important property of the binary material method is the way in which casting restrictions are

implemented. Without restrictions, all elements within the structure are “visible” and can therefore be added and

removed freely. Applying a casting restriction in terms of a (die/mold) pulling direction makes all but the elements

on the front and back surface (w.r.t. the casting direction) “invisible”, Fig. 1(b). This automatically prohibits the

formation of undercuts within the structure.

The methods and techniques presented in this paper are based on the restriction that the consideration of maximum

member sizes has to be used in conjunction with a casting restriction. In industrial applications, this does not limit

to the practical usability of the method, since the definition of a pulling direction can be seen as a prerequisite for

any optimization constraints which are in a sense more specialized.

5. A fast approximation of local structural dimensions

Knowledge about the local dimensions of the structure at every iteration step is a necessary prerequisite for any

member size constraints. Since the size of FE models for industrial applications in use today often exceed one

Million FE elements, computational performance is essential and effectively limits the affordable complexity of

the approximation technique. Also, the gathered member size information has to be accessible and compatible with

applied casting restrictions, which limit the direction in which the optimizer can affect the structure. Therefore, the

proposed member size approximation is carried out in two phases, which will be briefly explained below.

5.1. Determining the element depth

A straightforward approach for determining local member sizes (i.e. wall thicknesses) would be to determine the

minimal distance between opposing surface Finite Elements. Major drawbacks of this approach include the

nontrivial determination of opposing surface elements within a local region, as well as computing the distances for

a vast amount of element pairs. The idea behind a faster approach is the fact that, for regular voxel-based FE

meshes, local wall thicknesses correlate to the number of elements from the innermost element to its nearest

surface. The local wall thickness can then be roughly estimated as twice the maximum element depth (with an

inherent uncertainty of 1 voxel edge length). The accuracy of this approximation further depends on the search

radius construction needed to identify the nearest surface.

Using element neighbor definitions, one can define general and full neighbor relationships. The former share at

least one node, the latter share one element surface (or edge, for two-dimensional structures), as shown in Fig. 2(a).

If, starting from a source element, a search radius is constructed by alternating between those two neighbor types,

the approximation of a circle/sphere is considerably more accurate than by using only one neighbor type, Fig. 2(b).

general neighbors (top) – full neighbors (bottom)

(a) element neighbor definitions (b) search radius construction

alternating beween

neighbor definitions

using only

general neighbors

using only

full neighbors

Fig. 2: Search radius construction using element depth and element neighbor definitions

In an iterative procedure termed the onion shell loop, starting from the surface layer of the structure, each

following layer of elements is assigned incremental element depth values, until the innermost element layer is

reached. Each next layer is determined by alternating between the two above-mentioned neighbor definitions. This

procedure is equivalent to creating a search radius for each finite element within the structure to determine its

individual element depth separately, but is faster by two orders of magnitude. Calculating the element depth

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distribution of a structure with half a million Finite Elements takes less than 3 seconds on an average workstation.

Fig. 3 shows a colored representation of the resulting element depth distribution for a three-dimensional cantilever

beam structure.

element depth

1 2 3 4 5 6 7 8

Fig. 3: Colored representation of the element depth distribution within a cantilever beam model

5.2. Utilizing the casting projection surface

The maximum member size restriction is assumed to always be used in conjunction with a casting restriction.

Therefore, the information about the element depth distribution within the structure has to be accessible on the

front and back surface of the structure on an imaginary projection plane normal to the pull direction, Fig. 4(a).

Here, a projection ray is defined as a single row of finite elements parallel to the pull direction. By assigning the

highest element depth value within each row to all Finite Elements within the projection ray, the maximum

element depth information is accessible on the front and back surface of the structure. The resulting projected

element depth distributions for different casting directions are shown in Fig. 4(b).

projected element depth

1 2 3 4 5 6 7 8

pull

dir

ecti

on

section cut

pull

dir

ecti

on

pro

ject

ion r

ay

casting projectionprojection plane

design space

(a) Nomenclature regarding the casting projection concept (b) directional element depth projection

Fig. 4: The casting projection principle and its application to the cantilever beam shown in Fig. 3

By element depth projection, all Finite Elements within the structure are divided into groups with identical

projected element depths (termed thickness groups), which can then be operated on separately. Due to the fact that

each thickness group is a set of projection rays, which are by definition parallel to the pull direction, the formation

of undercuts can be avoided easily.

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6. A discrete maximum member size penalization technique

The implementation of member size restrictions in heuristic iterative optimization procedures is difficult. Two

main concerns lead to the decision to implement a discrete penalization technique to enforce structures which do

not exceed maximum member sizes.

First, the inclusion of a fixed member size restriction (a “hard limit”) proves to be challenging in a heuristic

environment. More importantly, the authors wanted to ensure the existence of a solution for every optimization

problem that was solvable by the unmodified optimization method without member size restrictions. One could

easily imagine imposing a member size limit that physically prohibits any resulting structure to support the defined

loading. Commercial optimization software like OptiStruct™ would either produce an unviable design proposal or

abort the optimization. By using a penalization scheme (i.e. a “soft limit”), element depths exceeding the defined

maximum could be avoided, as long as the structural integrity allowed for it. In theory, the design proposal would

converge into the unrestricted solution in the event of every bit of material being necessary and already in optimal

distribution, ensuring the existence of a solution rather than fulfilling the member size restrictions.

Second, the effectiveness of continuous penalization was investigated and turned out to be rather poor, since the

growth rule governing the optimizer (the hotspot correction mechanism HSC) was constantly reversing the

changes from the previous iteration induced by the penalization scheme, effectively slowing the optimization

progress to a halt, Fig. 5(a).

6.1. The element depth reduction event (EDR)

Rather than trying to penalize exceeding member sizes continuously, the authors propose a technique to reduce the

maximum element depth value to the user-defined allowable limit at discrete and recurring points during the

optimization process. The underlying heuristic is based on the idea to modify the topology noticeably in one step,

thus changing the flux of forces within and driving the optimizer to converge into a different topology with smaller

maximum member sizes, Fig. 5(b).

p phigh element depth

high element depthhigh element depth

low element depth

low element depth

low element depth

low element depth

solid

void

notch stress

added material

(a) continuous penalization: similar topology, high element depth

(b) discrete reduction event: changed topology, low element depth

HSC

HSC

substructure under

compressional loading p

HSC: hotspot correction mechanism

Fig. 5: Continuous penalization of high element depths vs. discrete element depth reduction events

Two subtractive mechanisms are proposed to reduce the maximum element depth to the allowable limit in one

step: shrink and split.

During shrinking elements of a thickness group are removed from the back, front or both surfaces inwards until the

allowable limit is reached, Fig. 6(a). Shrinking is therefore not applicable for structures that are two-dimensional

w.r.t. the pull direction. Depending on the stress levels of the front and back surface elements, either a one-sided

shrink (removing all elements from the side with the less stressed surface element) or a symmetric shrink is

performed (removing elements from both sides symmetrically).

During splitting all projection rays of elements from within the targeted thickness group are removed, splitting the

structure along said thickness group, Fig. 6(b).

6

pull

dir

ecti

on

..

..

..

..

shrink

..

..

..

..split

..

..

..

..

..

projection ray

..

..

..

..

projection ray

(a) shrink mechanism: removal of elements from the front and back of each projection ray

(b) split mechanism: removal of all projection rays within the targeted thickness group

element depth1 2 3

Fig. 6: Cross-sectional view of the splitting and shrinking mechanisms for an element depth limit of 2

Since the splitting mechanism potentially creates unconnected substructures, as demonstrated in Fig. 7(a), a

heuristic was developed to leave thin struts (adjacent projection rays) at highly stressed regions to connect both

adjacent thickness groups, Fig. 7(b). Through an efficient combination of shrinking and splitting, the reduction of

element depths of two- and three-dimensional structures down to the allowable limit in one EDR can be achieved

with relatively low amounts of damage to the structural integrity. As a trade-off, the reduction is not perfect. Some

localized regions, where connecting struts are kept, can still contain element depths above the allowable limit. This

does not have a negative effect, since the structural integrity has to be maintained. In fact, it was observed that

more material would have been added by the optimizer while repairing the structure, if no connecting struts were

left during the EDR event, leading to higher overall element depths.

(a) the “stamp-out-effect” of splitting (b) splitting with applied heuristics for connecting strut placement

almost fully

unconnected region

flux of forces

separated

splitsplit

connecting struts

forced

displ.

fixed

stress distribution (Mises) element depth distribution

element depths above limit

are shown in red

stress distribution (Mises)

Fig. 7: The “stamp-out-effect” of splitting is avoided by keeping connecting elements at highly stressed positions

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6.2. The discrete penalization cycle

After an EDR event, the structure is weakened, possibly to the point where it violates some or all of the

optimization constraints (e.g. displacement/stiffness/stress constraints). At this point, the aforementioned repair

capabilities of the optimizer come into play. The structure is repaired until all constraints are again fulfilled. After

the repair phase, the structure is allowed to be optimized without interference for a specifiable amount of iterations.

This so-called cooldown phase is necessary to remove excess material that was amassed during the repair phase,

and to “smooth” the structures flux of forces after an EDR event. A single EDR event is not sufficient for a

permanent reduction of element depths to the allowable limit, due to the repairs by the optimizer, which partially

reverse the changes induced by the EDR event. Thus, subsequent EDR events are necessary to further reduce the

element depths. The resulting cycle is termed the discrete penalization cycle.

For various academic problems the achievable average element depth converges after about 3-5 EDR events. To

test the efficiency of the penalization technique described in this paper, the popular cantilever beam problem was

chosen. The design space consists of 160x30x100 Finite Elements (young’s modulus 1000, poisson ratio 0.25,

edge length 1mm). The allowable element depth limit Dmax

was chosen to be 3, equating to a maximum member

size of roughly 6 mm. The structure is clamped on one side, and a vertical displacement of 5 mm is prescribed at

the center of the opposing free end. At the point of loading, a minimum reaction force of 4 kN is defined as an

optimization constraint.

EDR4MMS solution

EDR1

15

16

25

26

35

36

46

47

Projected element depth

EDR2 EDR3

Fig. 8: Development of the element depth distribution within a 3d cantilever beam example over the course of four

EDR events and subsequent repair phases

Fig. 8 shows the development of the element depth distribution over four EDR events. Here, the projected element

depths are color-coded in four groups of multiples of the allowable limit Dmax

. The numbers left of each structure

denote the iteration. The top right structure is termed the maximum member size solution and will hereinafter be

referred to as the MMS solution.

coo

ldo

wn

ph

ase

ED

R 2

ED

R 3

ED

R 4

ED

R 1

rep

air

ph

ase

226.925 226.271 222.875

160.000

210.000

260.000

310.000

360.000

410.000

460.000

0,50

0,70

0,90

1,10

1,30

1,50

1,70

1,90

2,10

2,30

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

num

ber

of

elem

ents

no

rmal

ized

const

rain

tval

ue

iteration φ φ_max Elementeelements

Fig. 9: Graph representation of the optimization progress and the discrete penalization cycle

8

The optimization progress is shown in Fig. 9. Here, the different phases within the penalization cycle can be easily

distinguished. After three repair phases, the optimization was stopped, due to sufficiently small changes in element

numbers to interpret the penalization efficiency. By comparing the MMS solution to a fully converged reference

solution without maximum member size restrictions, the overall element depth reduction can be related to the

amount of additional material necessary, Fig. 10. In other words, the local optimum with additional constraints is

worse than the less constrained reference solution, which is to be expected. On the other hand, by “investing” less

than 8% more material, the amount of elements above the allowable limit is reduced from 85% to less than 50%.

Additionally, the average element depth of those elements is drastically reduced.

reference solution after 100 iterations

amount of

additional material

needed for the

MMS solution:

+7,65%

MMS solution after 46 iterations

84.67% 49.82%

D ≤ Dmax

D ≤ 2xDmax

D ≤ 3xDmax

D > 3xDmax

Fig. 10: Element depth comparison of solutions obtained with and without maximum member size restriction

Since a two-dimensional representation is difficult to visualize, both the reference and MMS solution were

postprocessed by a smoothing algorithm, and are shown side by side in Fig. 11. The MMS solution shows a dual

upper and lower strut design, resembling an I-beam cross-section.

reference solution (green)

MMS solution (grey)

Fig. 11: Comparison of smoothed solutions with and without maximum member size restriction

Finally, the MMS solution was compared to a solution obtained by a mathematical optimization approach, namely

OptiStruct™, and termed the OS solution. Since the maximum member size restriction is a mathematical

constraint, i.e. a hard limit, in OptiStruct™, the allowable element depth limit was set to 8 rather than 6 as not to

9

favor the soft limit approach by the authors too much. The resulting design proposal is shown in Fig. 12. The OS

solution contains 45% more elements than the MMS solution and produces an infeasible design with a reaction

force of 2505 N, 37% less than required. Also, 4.6% of all element depths were still above the allowable limit of 4.

MMS solution: solid upper strut

OS solution: multiple scattered upper struts

0%

50%

100%

150% MMSOS

elements additional

material

elements

with D>Dmax

reaction

force

maxiumum member size restriction efficiency

OS solution: isometric view

Fig. 12: Direct comparison with a solution obtained by OptiStruct™

7. Branching and consecutive optimization paths

The introduction of EDR events allows for a straightforward realization of optimization branching. Before each

event, the structure is duplicated, one copy passes through unaltered, while the other is modified by the EDR event.

Through repetition, a binary tree of variations is generated, which differ in the amount of penalization events

performed upon them, Fig. 13. Each subtree follows its own discrete penalization cycle, resulting in an

asynchronous branching behavior. The specifiable maximum number of EDR events limits the number of possible

concurrent structural variants. Additionally, a repair time limit eliminates all structural variants that failed to be

repaired within a specifiable number of iterations to discard less promising structural variants.

reference sol.

0x EDR

2nd branching

variant 2

1x EDR

variant 1

2x EDR

unmodified

EDRrepair cooldown

cooldown unmodified

repair

unmodified

EDR

variant 3

repair time exceeded, discardedEDR

1st branching

2nd branching

Fig. 13: Principle of optimization branching, generating a binary tree of structural variants

In a two-dimensional version of the previously shown cantilever beam, a total of 17 variations were created with

enabled parallel optimization functionality. The reference solution and the three best variants are shown in Fig. 14.

Again, element depth values above the allowable limit are shown in red. The displayed performance value sets the

additionally needed amount of material in relation to the reduction of elements above the allowable limit. Across

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all variants, the percentage of elements above the allowable limit was reduced from 41% to 21%, while it was

reduced to 7% for the best variant. Analogously to the three-dimensional example, the necessary amount of

additional material was rather small, around 5% across the board.

EDR: 4

EDR: 5

EDR: 4

reference solution EDR: # of performed EDR events; performance value (higher is better)

EDR: 0

Fig. 14: Overview of concurrently optimized variants after 80 iterations and up to 5 EDR events

8. Conclusions

By direct comparison with a commercial topology optimization tool it was demonstrated that (for an isolated

example) a discrete maximum member size penalization technique can be combined with a heuristic iterative

optimization method and achieve promising and competitive results. Since the maximum member size limit is not

a hard limit at which casting defects will occur, the use of a soft limit approach can be beneficial. Considerable

reductions of element depths with the use of very little additional material were achieved. As a result, the

castability can be noticeably improved without gaining too much additional weight.

The introduction of an optimization branching mechanism allows for broader coverage of the solution space. The

element depth reduction performance between the generated variants differs noticeably, warranting further

investigation into optimal element depth reduction strategies. Concurrent optimizations will be a useful tool for

future in-depth investigations into the maximum member size penalization approach introduced in this paper.

9. References

[1] H. Eschenauer, N. Olhoff, Topology optimization of continuum structures: A review, Appl. Mech. Rev., 54

(4), 331-390, 2001.

[2] L. Harzheim and G. Graf, A review of optimization of cast parts using topology optimization II-Topology

optimization with manufacturing constraints, Struct. Multidisc. Optim., 31, 388-399, 2006.

[3] H. Thomas, M. Zhou and U. Schramm, Issues of commercial optimization software development, Struct.

Multidisc. Optim., 23, 97-110, 2002.

[4] M. Zhou, R. Fleury, Y.K. Shyy, H. Thomas and J.M. Brennan, Progress in topology optimization with

manufacturing constraints, American Institute of Aeronautics and Astronautics, Inc., 2002.

[5] FE-Design TOSCA.Structure, User’s manual. FE-Design GmbH, Karlsruhe, Germany, www.fe-design.de.

[6] Altair OptiStruct, Optistruct 11 User’s manual. Altair Engineering Inc., Troy, MI, www.altair.com.

[7] J. K. Guest, Imposing maximum length scale in topology optimization, Struct. Multidisc. Optim., 37,

463-473, 2009.

[8] S. Fiebig and J.K. Axmann: Intelligenter Leichtbau durch neue Topologieoptimierung für

Betriebsspannungen und plastisches Materialverhalten, 16. Kongress SIMVEC – Berechnung, Simulation

und Erprobung im Fahrzeugbau 2012, VDI-Berichte 2169, 695-712, November 20-21, Baden-Baden, 2012.

[9] S. Fiebig and J.K. Axmann, Combining nonlinear FEA simulations and manufacturing restrictions in a new

discrete Topology Optimization method, 9th World Congress on Structural and Multidisciplinary

Optimization, June 13 -17, Shizuoka, Japan, 2011.